The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon.
Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h Version of Finite Elements, see Paper.
We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.
16 June 2004.
SAM Report, ETH Zürich, N. 2004-05 .
M3AS: Mathematical Models and Methods in Applied Sciences, Vol. 15, n° 4, 2005, 575-622.
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